Statistical Sampling Techniques

• Statistical sampling concepts aid in making accurate inferences about populations via samples.
• Larger samples yield superior estimates with diminished variability.
• Confidence intervals offer a valuable range for estimating unknown population parameters.

Parameter and Statistic

• A parameter is a numerical value that describes a population characteristic, like population mean (µ) or standard deviation (σ).
• A statistic is a numerical measure computed from a sample, like sample mean (𝑥̄) or standard deviation (s), estimating population parameters.

Sampling Plan

• A sampling plan is a strategy for selecting a sample from a population to make inferences about the entire population.
• Simple Random Sampling (SRS) gives every element an equal chance of selection.
• Systematic Sampling involves selection following a fixed interval.
• Stratified Sampling divides a population into homogeneous subgroups.
• Cluster Sampling randomly selects clusters and includes all members.

Sampling Distribution

• A sampling distribution represents the probability distribution of a statistic from multiple random samples.
• It illustrates how a sample statistic behaves across repeated sampling.
• The mean of the sampling distribution equals the population mean.
• The distribution spread depends on the sample size and is indicated by the standard error.
• As sample size increases, the distribution becomes more normal due to the Central Limit Theorem (CLT).

Mean Sampling Distribution

• The sampling distribution of the mean is the probability distribution of sample means from repeated random samples.
• If the population distribution is normal, the sampling distribution of the mean is also normal.
• Even if the population isn't normal, the CLT ensures that with a larger sample size (n ≥ 30), the sampling distribution of the mean approximates a normal distribution.

Variance of a Sampling Distribution

• The variance of a sampling distribution measures the dispersion of sample statistics.
• As the sample size increases, the variance decreases, with sample means clustering more closely around the population mean.

Sampling Distribution of the Sample Mean

• This distribution describes the behavior of the sample mean across repeated samples.
• It follows a normal distribution if the population is normal.
• It approximates a normal distribution for large n (by CLT).
• The expected value (mean) equals the population mean.
• Standard error measures the variability of the sample mean.

Point Estimation

• A point estimate uses a single value to approximate a population parameter.
• Sample mean estimates the population mean.
• Sample proportion estimates the population proportion.

Interval Estimation

• An interval estimate gives a range of values where the population parameter is likely to lie, often using confidence intervals.

Peak Distribution

• In probability distribution, the peak indicates the most frequent values.
• In a normal distribution, the peak is at the mean (µ).
• A skewed distribution has a peak shifted left (negative skew) or right (positive skew).

Point and Confidence Estimation

• Point Estimation gives a best guess for a population parameter.
• Confidence Estimation provides a range (Confidence Interval or CI) where the population parameter is likely to be found.

Confidence Interval (CI)

• Computed as CI = 𝑥̄ \pm Z \times SE, where Z is the critical value and SE is the standard error.
• For example, a 95% CI for a sample mean of 50 with a standard error of 5 is (40.2, 59.8).

Length of Confidence Interval

• Confidence interval length is given by 2 \times Z \times SE.
• A larger sample size results in a shorter (more precise) confidence interval.
• A smaller sample size leads to a wider (less precise) confidence interval.

Determining Sample Size

• To achieve a specific confidence interval width, solve for n in the equation: n = \left(\frac{2Zσ}{W}\right)^2.